Week 9

You learn here how to count distinct orbits, and in the process
touch upon all the main themes of this book, going
the whole distance from diagnosing chaotic dynamics to
- while computing the topological
entropy from transition matrices/Markov graphs - our first
zeta function.
The long time dynamics of a discrete-time system is described by (1) generating all orbits by action of the 1-time step transition matrix, and (2) expressing the result as a "generating function". Here we derive the trace formula, the determinant and the topological zeta function. If you do not understand how to derive these, you'll be lost for the rest of the semester, and what fun is that?
Read sects. 18.1 - 18.4; 18.6 - 18.7, ChaosBook vers. approx 15.3.

Week 10

On the necessity of studying
the averages of observables in chaotic dynamics. Formulas
for averages are cast in a multiplicative form that
motivates the introduction of evolution operators.
Discrete-time and generating functions continued and compared. Infinitesimal time evolution related to the infinite time dynamics via a Laplace transform. The stage is set for the classical trace formula.
Read sects. 20.1 and 20.2. Skip sect. 20.1.3 "Moments, cumulants"

Week 11

If there is one idea that one should learn about chaotic dynamics, it happens
in this chapter: the (global) spectrum of the evolution is dual to the
(local) spectrum of periodic orbits. The duality is made precise by
means of trace formulas. The course is OVER. Trace formulae are beautiful, and there is nothing more to say. Just some mopping up to do.

We derive the spectral determinants, dynamical zeta functions.
While traces and determinants
are formally equivalent, determinants are the tool of choice when it comes
to computing spectra. Skip sects. 20.5 and 20.6.

Week 13

Read sect. 28.2: A class of simple 1-dimensional dynamical systems where all transport coefficients can be evaluated analytically,
by hand. Diffusion is a non-monotonic function of the local expansion rate, and it is non-gaussian, with a non-vanishing kurtosis.
Perhaps the most fundamental diagnostic of deterministic chaos is the non-differentiable dependence of its transport coefficients on smooth variations of system parameters.

Finite groups. Cyclic groups of two and three elements. Symmetries of a triangle, six element group multiplication table. Matrix representations. Regular representations. This is standard material, not written up in ChaosBook,
but necessary for the course.
We liked Tinkham, chapter 2,
or Dresselhaus chapters 1 and 2 (early version available as lecture notes).

Week 14

Characters. Orthogornality relations. Character tables. If you have understood character projection operators, we are set.
We liked Tinkham, chapter 3,
or Dresselhaus chapters 3 and 4 (early version available as lecture notes).

Symmetries simplify and improve the cycle expansions in a rather beautiful, not entirely obvious way, by factorizing cycle expansions. Read sects. 25.1 and 25.2. For a one-d map with reflection symmetry determinants factorize into symmetric and and antisymmetric ones, and each one receives contributions from all kinds of orbits. In a not entirely obvious way. A triple home run: simpler symbolic dynamics, fewer cycles needed, much better convergence of cycle expansions. Once you master this, going back is unthinkable.

Week 15

3 disk pinball symmetries suffice to illustrate all that is needed to factorized spectral determinants for any system with a discrete symmetry: character. Discrete symmetry tiles the state space, and dynamics can be reduced to dynamics on the fundamental domain, together with a finite matrix that keeps track of the tile the full state space trajectory lands on. We need some group theory (one needs to underatand the projection to irreducible representations) and illustrate how different classes of periodic orbits contribute to different invariant subspaces for the 3-disk pinball. Read sects. 25.2 - 25.6.

Trace formulas relate short time dynamics (unstable period
ic orbits) to long time invariant state
space densities (natural measure).
A trace formula for a partially hyperbolic (N+ 1)-dimensional compact manifold invariant under a
global continuous symmetry is derived. In this extension of “periodic orbit” theory there are
no or very few periodic orbits - the relative periodic orbits
that the trace formula has support on
are almost never eventually periodic

Week 16

Flows described by PDEs are said to be `infinite dimensional' because if one writes them down as a set of ODEs, one needs infinitely many of them to represent the dynamics of one PDE. The long-time dynamics of many such systems of physical interest is finite-dimensional. Here we cure you of the fear of infinite-dimensional flows.

In the world of everyday, moderately turbulent fluids flowing across planes and down pipes, a velvet revolution is taking place. Experiments are as detailed as simulations, there is a zoo of exact numerical solutions that one dared not dream about a decade ago, and portraits of turbulent fluid's state space geometry are unexpectedly elegant.
We take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the tutorial is aimed at anyone who had ever wondered how we know a cloud when we see one, if no cloud is ever seen twice? And how do we turn that into mathematics?